Ulrich Bauer

I am an assistant professor in the Geometry & Visualization group at TUM. I did my PhD with Max Wardetzky in the research group Discrete Differential Geometry at the University of Göttingen, and a postdoc in Herbert Edelsbrunner's research group at IST Austria.

Research interests:

Discrete/Computational Geometry/Topology, Algorithms, Combinatorics.

Contact:
 Prof. Dr. Ulrich Bauer Geometrie & Visualisierung (M10) Fakultät für Mathematik Technische Universität München Boltzmannstraße 3 D-85747 Garching bei München Vox: +49 89 289 18361 Net: mail (at) ulrich-bauer.org

Preprints
 The Reeb Graph Edit Distance is Universal Ulrich Bauer, Claudia Landi, and Facundo Mémoli. Abstract: We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specific construction, we show that the interleaving distance and the functional distortion distance on Reeb graphs are not universal. [arXiv] Hardness of Approximation for Morse Matching Ulrich Bauer and Abhishek Rathod. Abstract: We consider the approximability of maximization and minimization variants of the Morse matching problem, posed as open problems by Joswig and Pfetsch. We establish hardness results for Max-Morse matching and Min-Morse matching. In particular, we show that, for a simplicial complex with n simplices and dimension $d\leq 3$, it is NP-hard to approximate Min-Morse matching within a factor of $O (n^{1-\epsilon})$, for any $\epsilon> 0$. Moreover, using an L-reduction from Degree 3 Max-Acyclic Subgraph to Max-Morse matching, we show that it is both NP-hard and UGC-hard to approximate Max-Morse matching for simplicial complexes of dimension $d\leq 2$ within certain explicit constant factors. [arXiv] Persistent Betti numbers of random Čech complexes Ulrich Bauer and Florian Pausinger. Abstract: We study the persistent homology of random Čech complexes. Generalizing a method of Penrose for studying random geometric graphs, we first describe an appropriate theoretical framework in which we can state and address our main questions. Then we define the kth persistent Betti number of a random Čechcomplex and determine its asymptotic order in the subcritical regime. This extends a result of Kahle on the asymptotic order of the ordinary kth Betti number of such complexes to the persistent setting. [arXiv] Persistence in sampled dynamical systems faster Ulrich Bauer, Herbert Edelsbrunner, Grzegorz Jablonski, and Marian Mrozek. Abstract: We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior, and to recover the eigenspaces of the endomorphism on homology induced by the self-map. The chain maps are constructed using discrete Morse theory for Cech and Delaunay complexes, representing the requisite discrete gradient field implicitly in order to get fast algorithms. [arXiv] Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem Ulrich Bauer and Michael Lesnick. Abstract: Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of real intervals. Recent work of Edelsbrunner, Jablonski, and Mrozek suggests an equivalent description of barcodes as functors R-> Mch, where R is the poset category of real numbers and Mch is the category whose objects are sets and whose morphisms are matchings (ie, partial injective functions). Such functors form a category Mch^ R whose morphisms are the natural transformations. Thus, this interpretation of barcodes gives us a hitherto unstudied categorical structure on barcodes. The aim of this note is to show that this categorical structure leads to surprisingly simple reformulations of both the well-known stability theorem for persistent homology and a recent generalization called the induced matching theorem. [arXiv]

Software
 Ripser: a lean C++ code for the computation of Vietoris–Rips persistence barcodes Ulrich Bauer (2015–2018). Licensed under LGPL. Description: Ripser is a lean C++ code for the computation of Vietoris–Rips persistence barcodes. It can do just this one thing, but does it extremely well. The implementation is based on an implicit algorithmic representation of the coboundary operator and of the filtration order, avoiding the explicit construction of the filtration boundary matrix. Our software shows significant improvements over existing software both in time and memory usage. The source code consists of less than 1000 lines of C++ code. It supports for coefficients in prime finite fields and sparse distance matrices for computing the Vietoris–Rips persistence barcodes only up to a specified threshold. No external dependencies are required. To see a live demo of Ripser's capabilities, go to live.ripser.org. The computation happens inside the browser (using PNaCl on Chrome and JavaScript via Emscripten on other browsers). [live] [github] PHAT (Persistent Homology Algorithm Toolbox) Ulrich Bauer, Michael Kerber, Jan Reininghaus. Copyright IST Austria (2013–2017). Licensed under LGPL. Description: PHAT is an open-source C++ library for the computation of persistent homology by matrix reduction, targeted towards developers of software for topological data analysis. We aim for a simple generic design that decouples algorithms from data structures without sacrificing efficiency or user-friendliness. PHAT provides numerous different reduction strategies as well as data types to store and manipulate the boundary matrix. [bitbucket]

Publications